A Fractional Advection–Dispersion Framework with Non-Uniform Clay Adsorption Fields for Enhanced Reservoir History Matching and Fractal Validation

History matching traditionally assumes a spatially uniform clay adsorption coefficient Kd, overlooking the fractal heterogeneity of clay mineral distributions in real reservoirs. This study generalizes the classical advection–dispersion framework by treating Kd as a non-uniform field whose logarithmic fluctuations follow fractional Brownian motion (fBm) statistics with Hurst exponent H, and by coupling the resulting heterogeneous retardation to a fractional advection–dispersion equation through the continuous-time random walk bridge β = 2/(2H + 1). A Karhunen–Loève (K-L) expansion compresses the field into a small set of spectral modes, and an adjoint-gradient inversion recovers the clay field from tracer data. The main operational limitation of the K-L approach is that sub-grid variability below the retained spectral cutoff is smoothed; within this boundary the inversion is efficient and numerically stable. Two independent validation paths are pursued. First, a cross-fitting experiment shows that when the true physics is sub-diffusive (β = 0.85), the classical advection–dispersion equation yields a negative coefficient of determination (R2 = −0.927), whereas the correctly specified fractional model fits the data, establishing fractional derivatives as physically necessary rather than mathematically optional. Second, the publicly available EGS Collab DNA-tracer dataset independently confirms statistically significant clay adsorption (retardation R = 1.250 ± 0.009, p < 0.0001). The recovered clay field attains pixel-scale accuracy R2η = 0.844 (Spearman ρ = 0.923) and R2 > 0.94 at reservoir-relevant scales of five cells or coarser. These results indicate that fractal-aware clay-field inversion provides a physically verifiable, rather than purely empirical, pathway for history matching, with direct implications for field-scale contaminant transport prediction and environmental subsurface management, and the evaluation of flue gas enhanced thermal recovery in thin-layer heavy oil reservoirs.